Predicted values for varying values of X2 are identical when X1 = C, because predicted values fall at a single point for any value of X2. We altered Equation 1 by replacing X1 with (X1 − C) and placing the new intercept (Equation 9) in the equation. In this model, B2 becomes inestimable, because X2 has no relation to Y at the cross-over point on X1. The re-parameterized equation thus becomes: (10)Yi=A0+B1(X1i−C)+B3((X1i−C)·X2i)+Ei where all symbols were defined previously. Equation 10 is a four-parameter equation because C is now a parameter to be estimated, with the same number of free parameters as Equations 1 and 2. Symbols for B1 and B3 remain the same as in Equation 1 because these coefficients are unchanged by re-centering X1 at C. Equation 10 is a re-parameterization of Equation 1 (as shown in supplemental material2) and thus leads to identical predicted values when plotting interactions. We also note that, because of its form, Equation 10 must be estimated using a non-linear regression program, rather than a standard linear regression program.2
